Chapter 11: Problem 10
[Computer] In general, the analysis of coupled oscillators with dissipative forces is much more complicated than the conservative case considered in this chapter. However, there are a few cases where the same methods still work, as the following problem illustrates: (a) Write down the equations of motion corresponding to (11.2) for the equal-mass carts of Section 11.2 with three identical springs, but with each cart subject to a linear resistive force \(-b \mathbf{v}\) (same coefficient \(b\) for both carts). (b) Show that if you change variables to the normal coordinates \(\xi_{1}=\frac{1}{2}\left(x_{1}+x_{2}\right)\) and \(\xi_{2}=\frac{1}{2}\left(x_{1}-x_{2}\right),\) the equations of motion for \(\xi_{1}\) and \(\xi_{2}\) are uncoupled. (c) Write down the general solutions for the normal coordinates and hence for \(x_{1}\) and \(x_{2}\). (Assume that \(b\) is small, so that the oscillations are underdamped.) (d) Find \(x_{1}(t)\) and \(x_{2}(t)\) for the initial conditions \(x_{1}(0)=A\) and \(x_{2}(0)=v_{1}(0)=v_{2}(0)=0,\) and plot them for \(0 \leq t \leq 10 \pi\) using the values \(A=k=m=1,\) and \(b=0.1\).
Short Answer
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Key Concepts
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